A Squarefree Upper Bound for Coprime Adjacent Divisors

We let n be squarefree with ω(n) = k, and let 1 = d_1 < d_2 < ... < d_τ(n) = n be the increasing sequence of divisors of n. Define τ_⊥(n) to be the number of adjacent divisor pairs (d_i, d_{i+1}) such that gcd(d_i, d_{i+1}) = 1. We prove the squarefree upper bound τ_⊥(n) ≤ 2 · sum_{j=0}^{t} C(k,j) + sum_{j=0}^{k-2t-2} C(k,j) for every t ≥ 0, with the convention that an empty binomial sum is zero. Optimizing at t ~ k/3 gives τ_⊥(n) ≪ k (3/2^{2/3})^k = k (1.88988...)^k. Thus the squarefree extremal function g_sf(k) = max_{ω(n)=k, n squarefree} τ_⊥(n) satisfies g_sf(k) ≤ (1.88988... + o(1))^k. This improves the elementary reciprocity bound g_sf(k) ≤ 2^{k-1}. (This is a companion to A Reduction of the Squarefree Coprime Adjacent Divisor Problem and a Conditional Golden-Ratio Lower Bound.)